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Tauberian theorems for generalized functions in the scale of regularly varying functions and functionals. (Russian) Zbl 1037.46040
The authors give in the series of their papers, of which the present paper is a part, necessary and sufficient conditions for an \(f\in\mathcal S_+'\) to be quasiasymptotically bounded at infinity (zero) in terms of the behaviour of its Laplace transform at zero (infinity). Here the kernel of the Laplace transform is changed by an \(\omega\in\mathcal S\) with the integral different from zero and the regularization of \(f\): \(L_f^\omega(x,y)=\langle f,\frac1y\omega (\frac{\cdot-x} y)\rangle\) is considered. Vector valued and, in particular, Banach valued (B-valued) tempered distributions are also considered. With \(\rho\) being a regularly varying function, \(\rho(k)f(\cdot/k)\in\mathcal S_+'\) is B-valued if and only if \(\rho(k)\| y^b L_f^\omega(x/k,y/k)\|\) satisfies appropriate assumptions. Several results of this type, related to the shift asymptotics, are also given. The interesting and new concept of a regularly varying functional is introduced and in terms of this notion appropriate Abel and Tauber-type theorems are given. The results are applied to a heat equation with Cauchy data in \(\mathcal S_+'\).

46F12 Integral transforms in distribution spaces
44A10 Laplace transform
40E05 Tauberian theorems
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